Lesson 15 of 17
The big picture
Show on a big screenEvery build and tiling idea from this track fits on one screen.
| Idea | Remember | Example |
|---|---|---|
| Copy a corner | Same radius. Carry the gap. | A copy of 62° is 62°. |
| The perpendicular bisector | Equal arcs from both ends cross twice. | It meets the segment at 90°. |
| Split a corner | Equal marks give equal halves. | 62° splits into two 31° corners. |
| A parallel | Copy the corner a crossing line makes. | Same gap at two places. |
| Three lengths | Base, arc, arc, join. | 3, 4, 5 closes. 3, 4, 9 does not. |
| Two lengths, angle between | The angle must sit between them. | Move it away, and sometimes two triangles fit. |
| A tiling | No gaps and no overlaps. | 5 × 5 has 25 squares. 2 × 1 tiles cannot cover it. |
| The whole turn | Tile corners at one point add to 360°. | 108° × 3 = 324°, so 36° short. |
| A square line | To draw a perpendicular, you need a segment on the line. Its perpendicular bisector must go through your point. On the line, make the point the middle of the segment. Off the line, pick ends that are equally far from it. | 26 mm above a line: the span must be over 26 mm. |
| 60°, 30° and 15° | Two arcs of one span make an equilateral triangle. Its corner is 60°. Bisect a corner to halve it. Bisect 60° to get 30°. Bisect 30° to get 15°. | Bisect 60° twice for 15°. |
| A hexagon | Draw a circle. Keep the compass at the radius. Step it round the circle. Six steps come back to the start. Join the six points. That is a regular hexagon. | A 30 mm radius gives 30 mm sides. |
| The tangram | A tangram is a square cut into seven pieces. Two big triangles are 1/4 each. The medium triangle is 1/8. So are the little square and the slanted piece. Two small triangles are 1/16 each. The shares add to 1. | 2 of 16 small triangles is 1/8. |
| Tiling the whole floor | Use copies of one regular shape, edge to edge. They tile when whole corners fill 360°. Divide 360° by the corner. A whole number means they fit. Only the triangle, square and hexagon do. | 360 ÷ 120 = 3. 360 ÷ 108 is not whole. |
Chapter 6 of Part II is Constructions and Tilings. Your book asks you to build many designs. Try them with a compass.
One habit runs through all of them. Check a construction by measuring it.